For \( b=1 \) , the famous Gap-Labelling theorem [1]} states that for any spectral gap \(G\) , there exists a unique \(k\in {\mathbb {Z}}^d\)  such that \(2\rho \equiv \langle k, \alpha \rangle \mod {{\mathbb {Z}}}\) , where \( \rho \)  is its fibered rotation number (consult section REF ). That is, all the spectral gaps can be labelled by integer vectors: we
denote by \(G_k(v ) = (E^{-}_k , E^{+}_k )\)  the gap with label \(k\ne 0\) .
When \(E^{-}_k=E^+_k\) , we
say the gap is collapsed.
We also set \(E_0^{-}:= \inf \Sigma _{v,\alpha },E_0^+ := \sup \Sigma _{v,\alpha },\)  and
\(G_0(v ) := (-\infty , E_0^{-}) \cup (E_0^+,\infty )\) . The consideration of the estimates of gaps, may go back to a question that whether the spectrum is a Cantor set for the almost Mathieu operator (AMO), for which \( v(\cdot )=2\lambda \cos 2\pi (\cdot ) \) , \( \lambda \ne 1 \)  and \( \alpha \in {\mathbb {R}}\backslash {\mathbb {Q}}\) . This is so-called Ten Martini Problem stated by Barry Simon after an offer of Mark Kac in 1981 [2]}. After works of many mathematicians, this problem was finally solved by Avila-Jitomirskaya [3]}. The Dry Ten Martini Problem further conjectures that all gaps \( G_k \)  labeled by the Gap-Labelling theorem are open, for AMO case. This problem was partially resolved in [4]}, [3]}, [6]}, [7]}. Moreover, Leguil-You-Zhao-Zhou [8]} obtained the exponential asymptotic behavior on the gaps of noncritical AMO recently, which gives a quantitative estimate on their size.
